Nine-point circle and how to construct it in Geogebra

There are quite a few circles associated with any given triangle. A circumcircle for example, is the name of the circle that passes through the three vertices of a triangle. 

There is also an incircle which is the circle of the largest possible radius that can fit inside a triangle.

Another one is the nine-point circle. It is the circle that passes through the nine points in the plane of a triangle, hence the name. These nine points are : 

• Feet of the three altitudes(an altitude is a segment from a vertex perpendicular to a line containing the side opposite to the vertex)

• Midpoints of the three sides of triangle(also said to be the points where the three medians meet the sides)

• Midpoints of the three line segments from the vertices to the orthocenter - the point where the three altitudes of a triangle intersect. 

In other words, these nine points are concyclic for any given triangle. Meaning, for a given triangle they lie on the circumference of the circle - the nine-point circle. Its center is called the nine-point center. 

In the figure above, in △ABC, nine concyclic points are : 

P, Q & R   ⇒        Feet of the three altitudes

X, Y & Z        ⇒     Midpoints of the sides(the medians are not shown)

M, N & O     ⇒     Midpoints of the segments from the vertices A, B and C to the  

                                 Orthocenter H. 

The big blue dot inside the triangle is to hide the fact that my measurements were off by a degree or two, I couldn’t get the altitudes to intersect at a single point. But right below this para there is another figure which is perfect in all measures because i used an online graphing calculator to make it. This one is an example of the nine-point circle for an obtuse angled triangle. This one i made in Geogebra - an online graphing calculator.

You could try and construct one by yourself using any graphing calculator online. It’s simple, even for those who have never used a graphing calculator before. Click here for a step by step guide on how to construct the nine-point circle of a triangle in Geogebra.

There are tons of properties of the nine-point circle(and its center). You will find many of them mentioned here if not all.